# 2-torsion in the n-solvable filtration of the knot concordance group

## TL;DR

This paper demonstrates that for the n-solvable filtration of the knot concordance group, the associated graded groups contain infinitely many elements of order 2, with specific properties and distinctions based on classical invariants.

## Contribution

It proves the existence of infinitely many 2-torsion elements in the graded groups of the n-solvable filtration, expanding prior knowledge from lower levels to higher levels.

## Key findings

- Each G_n contains infinite 2-torsion elements for n>1.
- These elements are represented by negative amphichiral knots with trivial classical invariants.
- The knots are distinguished by their Alexander polynomials and higher-order Alexander modules.

## Abstract

In 1997 Cochran-Orr-Teichner introduced a natural filtration, called the n-solvable filtration, of the smooth knot concordance group, C. Its terms {F_n} are indexed by half integers. We show that each associated graded abelian group G_n=F_n/F_{n.5}, n>1, contains infinite linearly independent sets of elements of order 2 (this was known previously for n=0,1). Each of the representative knots is negative amphichiral, with vanishing s-invariant, tau-invariant, delta-invariants and Casson-Gordon invariants. Moreover each is smoothly slice in a rational homology 4-ball. In fact we show that there are many distinct such classes in G_n, distinguished by their classical Alexander polynomials and by the orders of elements in their higher-order Alexander modules.